LMFDB - Embedded newform 1024.4.a.n.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1024,4,Mod(1,1024)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1024.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1024 = 2^{10} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1024.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$60.4179558459$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 36x^{8} + 405x^{6} - 1380x^{4} + 420x^{2} - 32 $ x^10 - 36x^8 + 405x^6 - 1380x^4 + 420x^2 - 32
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	no (minimal twist has level 16)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.357936$ of defining polynomial
Character	$\chi$	$=$	1024.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.77277 q^{3} +6.59550 q^{5} +24.8965 q^{7} +33.4160 q^{9} +O(q^{10})$	$q+7.77277 q^{3} +6.59550 q^{5} +24.8965 q^{7} +33.4160 q^{9} +31.5979 q^{11} +15.9402 q^{13} +51.2653 q^{15} +88.4846 q^{17} -53.4838 q^{19} +193.515 q^{21} +48.1224 q^{23} -81.4994 q^{25} +49.8702 q^{27} +14.7689 q^{29} -96.9578 q^{31} +245.604 q^{33} +164.205 q^{35} +230.911 q^{37} +123.899 q^{39} -360.519 q^{41} -141.774 q^{43} +220.395 q^{45} -220.669 q^{47} +276.837 q^{49} +687.771 q^{51} -248.551 q^{53} +208.404 q^{55} -415.717 q^{57} -572.767 q^{59} +939.852 q^{61} +831.942 q^{63} +105.133 q^{65} -151.854 q^{67} +374.045 q^{69} -215.050 q^{71} -668.587 q^{73} -633.476 q^{75} +786.679 q^{77} +822.956 q^{79} -514.603 q^{81} -462.269 q^{83} +583.600 q^{85} +114.795 q^{87} +262.733 q^{89} +396.855 q^{91} -753.631 q^{93} -352.752 q^{95} -150.801 q^{97} +1055.88 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 28 q^{7} + 54 q^{9}+O(q^{10}) $ 10 * q + 28 * q^7 + 54 * q^9	$ 10 q + 28 q^{7} + 54 q^{9} + 124 q^{15} + 4 q^{17} + 276 q^{23} + 50 q^{25} + 368 q^{31} - 4 q^{33} + 732 q^{39} + 944 q^{47} - 94 q^{49} + 1380 q^{55} + 108 q^{57} + 2628 q^{63} - 492 q^{65} + 3468 q^{71} - 296 q^{73} + 4416 q^{79} - 482 q^{81} + 6036 q^{87} + 88 q^{89} + 6900 q^{95} - 4 q^{97}+O(q^{100}) $ 10 * q + 28 * q^7 + 54 * q^9 + 124 * q^15 + 4 * q^17 + 276 * q^23 + 50 * q^25 + 368 * q^31 - 4 * q^33 + 732 * q^39 + 944 * q^47 - 94 * q^49 + 1380 * q^55 + 108 * q^57 + 2628 * q^63 - 492 * q^65 + 3468 * q^71 - 296 * q^73 + 4416 * q^79 - 482 * q^81 + 6036 * q^87 + 88 * q^89 + 6900 * q^95 - 4 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	7.77277	1.49587	0.747935	−	0.663771i	$-0.231045\pi$
$3$	7.77277	1.49587	0.747935	+	0.663771i	$0.231045\pi$
$4$	0	0
$5$	6.59550	0.589919	0.294960	−	0.955510i	$-0.404694\pi$
$5$	6.59550	0.589919	0.294960	+	0.955510i	$0.404694\pi$
$6$	0	0
$7$	24.8965	1.34429	0.672143	−	0.740422i	$-0.265374\pi$
$7$	24.8965	1.34429	0.672143	+	0.740422i	$0.265374\pi$
$8$	0	0
$9$	33.4160	1.23763
$10$	0	0
$11$	31.5979	0.866103	0.433052	−	0.901369i	$-0.357437\pi$
$11$	31.5979	0.866103	0.433052	+	0.901369i	$0.357437\pi$
$12$	0	0
$13$	15.9402	0.340078	0.170039	−	0.985437i	$-0.445611\pi$
$13$	15.9402	0.340078	0.170039	+	0.985437i	$0.445611\pi$
$14$	0	0
$15$	51.2653	0.882443
$16$	0	0
$17$	88.4846	1.26239	0.631196	−	0.775623i	$-0.282564\pi$
$17$	88.4846	1.26239	0.631196	+	0.775623i	$0.282564\pi$
$18$	0	0
$19$	−53.4838	−0.645790	−0.322895	−	0.946435i	$-0.604656\pi$
$19$	−53.4838	−0.645790	−0.322895	+	0.946435i	$0.604656\pi$
$20$	0	0
$21$	193.515	2.01088
$22$	0	0
$23$	48.1224	0.436270	0.218135	−	0.975919i	$-0.430003\pi$
$23$	48.1224	0.436270	0.218135	+	0.975919i	$0.430003\pi$
$24$	0	0
$25$	−81.4994	−0.651995
$26$	0	0
$27$	49.8702	0.355464
$28$	0	0
$29$	14.7689	0.0945692	0.0472846	−	0.998881i	$-0.484943\pi$
$29$	14.7689	0.0945692	0.0472846	+	0.998881i	$0.484943\pi$
$30$	0	0
$31$	−96.9578	−0.561746	−0.280873	−	0.959745i	$-0.590624\pi$
$31$	−96.9578	−0.561746	−0.280873	+	0.959745i	$0.590624\pi$
$32$	0	0
$33$	245.604	1.29558
$34$	0	0
$35$	164.205	0.793020
$36$	0	0
$37$	230.911	1.02599	0.512994	−	0.858392i	$-0.328536\pi$
$37$	230.911	1.02599	0.512994	+	0.858392i	$0.328536\pi$
$38$	0	0
$39$	123.899	0.508713
$40$	0	0
$41$	−360.519	−1.37326	−0.686629	−	0.727008i	$-0.740910\pi$
$41$	−360.519	−1.37326	−0.686629	+	0.727008i	$0.740910\pi$
$42$	0	0
$43$	−141.774	−0.502797	−0.251398	−	0.967884i	$-0.580890\pi$
$43$	−141.774	−0.502797	−0.251398	+	0.967884i	$0.580890\pi$
$44$	0	0
$45$	220.395	0.730102
$46$	0	0
$47$	−220.669	−0.684849	−0.342425	−	0.939545i	$-0.611248\pi$
$47$	−220.669	−0.684849	−0.342425	+	0.939545i	$0.611248\pi$
$48$	0	0
$49$	276.837	0.807104
$50$	0	0
$51$	687.771	1.88838
$52$	0	0
$53$	−248.551	−0.644172	−0.322086	−	0.946710i	$-0.604384\pi$
$53$	−248.551	−0.644172	−0.322086	+	0.946710i	$0.604384\pi$
$54$	0	0
$55$	208.404	0.510931
$56$	0	0
$57$	−415.717	−0.966019
$58$	0	0
$59$	−572.767	−1.26386	−0.631932	−	0.775024i	$-0.717738\pi$
$59$	−572.767	−1.26386	−0.631932	+	0.775024i	$0.717738\pi$
$60$	0	0
$61$	939.852	1.97272	0.986358	−	0.164614i	$-0.0526377\pi$
$61$	939.852	1.97272	0.986358	+	0.164614i	$0.0526377\pi$
$62$	0	0
$63$	831.942	1.66373
$64$	0	0
$65$	105.133	0.200619
$66$	0	0
$67$	−151.854	−0.276894	−0.138447	−	0.990370i	$-0.544211\pi$
$67$	−151.854	−0.276894	−0.138447	+	0.990370i	$0.544211\pi$
$68$	0	0
$69$	374.045	0.652604
$70$	0	0
$71$	−215.050	−0.359461	−0.179731	−	0.983716i	$-0.557523\pi$
$71$	−215.050	−0.359461	−0.179731	+	0.983716i	$0.557523\pi$
$72$	0	0
$73$	−668.587	−1.07195	−0.535974	−	0.844235i	$-0.680055\pi$
$73$	−668.587	−1.07195	−0.535974	+	0.844235i	$0.680055\pi$
$74$	0	0
$75$	−633.476	−0.975300
$76$	0	0
$77$	786.679	1.16429
$78$	0	0
$79$	822.956	1.17202	0.586012	−	0.810303i	$-0.300697\pi$
$79$	822.956	1.17202	0.586012	+	0.810303i	$0.300697\pi$
$80$	0	0
$81$	−514.603	−0.705902
$82$	0	0
$83$	−462.269	−0.611332	−0.305666	−	0.952139i	$-0.598879\pi$
$83$	−462.269	−0.611332	−0.305666	+	0.952139i	$0.598879\pi$
$84$	0	0
$85$	583.600	0.744710
$86$	0	0
$87$	114.795	0.141463
$88$	0	0
$89$	262.733	0.312918	0.156459	−	0.987684i	$-0.449992\pi$
$89$	262.733	0.312918	0.156459	+	0.987684i	$0.449992\pi$
$90$	0	0
$91$	396.855	0.457162
$92$	0	0
$93$	−753.631	−0.840300
$94$	0	0
$95$	−352.752	−0.380964
$96$	0	0
$97$	−150.801	−0.157850	−0.0789251	−	0.996881i	$-0.525149\pi$
$97$	−150.801	−0.157850	−0.0789251	+	0.996881i	$0.525149\pi$
$98$	0	0
$99$	1055.88	1.07192
$100$	0	0
$101$	−690.115	−0.679891	−0.339945	−	0.940445i	$-0.610409\pi$
$101$	−690.115	−0.679891	−0.339945	+	0.940445i	$0.610409\pi$
$102$	0	0
$103$	1840.58	1.76075	0.880377	−	0.474275i	$-0.157290\pi$
$103$	1840.58	1.76075	0.880377	+	0.474275i	$0.157290\pi$
$104$	0	0
$105$	1276.33	1.18626
$106$	0	0
$107$	−112.302	−0.101464	−0.0507322	−	0.998712i	$-0.516155\pi$
$107$	−112.302	−0.101464	−0.0507322	+	0.998712i	$0.516155\pi$
$108$	0	0
$109$	−1347.72	−1.18429	−0.592146	−	0.805831i	$-0.701719\pi$
$109$	−1347.72	−1.18429	−0.592146	+	0.805831i	$0.701719\pi$
$110$	0	0
$111$	1794.82	1.53475
$112$	0	0
$113$	720.469	0.599788	0.299894	−	0.953973i	$-0.403049\pi$
$113$	720.469	0.599788	0.299894	+	0.953973i	$0.403049\pi$
$114$	0	0
$115$	317.391	0.257364
$116$	0	0
$117$	532.657	0.420890
$118$	0	0
$119$	2202.96	1.69702
$120$	0	0
$121$	−332.571	−0.249865
$122$	0	0
$123$	−2802.23	−2.05422
$124$	0	0
$125$	−1361.97	−0.974544
$126$	0	0
$127$	2622.35	1.83225	0.916124	−	0.400895i	$-0.131301\pi$
$127$	2622.35	1.83225	0.916124	+	0.400895i	$0.131301\pi$
$128$	0	0
$129$	−1101.97	−0.752119
$130$	0	0
$131$	929.950	0.620230	0.310115	−	0.950699i	$-0.399632\pi$
$131$	929.950	0.620230	0.310115	+	0.950699i	$0.399632\pi$
$132$	0	0
$133$	−1331.56	−0.868127
$134$	0	0
$135$	328.919	0.209695
$136$	0	0
$137$	2511.52	1.56623	0.783117	−	0.621874i	$-0.213629\pi$
$137$	2511.52	1.56623	0.783117	+	0.621874i	$0.213629\pi$
$138$	0	0
$139$	−1535.86	−0.937195	−0.468598	−	0.883412i	$-0.655241\pi$
$139$	−1535.86	−0.937195	−0.468598	+	0.883412i	$0.655241\pi$
$140$	0	0
$141$	−1715.21	−1.02445
$142$	0	0
$143$	503.677	0.294543
$144$	0	0
$145$	97.4080	0.0557882
$146$	0	0
$147$	2151.79	1.20732
$148$	0	0
$149$	−3230.96	−1.77645	−0.888223	−	0.459413i	$-0.848060\pi$
$149$	−3230.96	−1.77645	−0.888223	+	0.459413i	$0.848060\pi$
$150$	0	0
$151$	−2814.39	−1.51677	−0.758383	−	0.651809i	$-0.774010\pi$
$151$	−2814.39	−1.51677	−0.758383	+	0.651809i	$0.774010\pi$
$152$	0	0
$153$	2956.80	1.56237
$154$	0	0
$155$	−639.485	−0.331385
$156$	0	0
$157$	−1281.71	−0.651541	−0.325770	−	0.945449i	$-0.605624\pi$
$157$	−1281.71	−0.651541	−0.325770	+	0.945449i	$0.605624\pi$
$158$	0	0
$159$	−1931.93	−0.963598
$160$	0	0
$161$	1198.08	0.586472
$162$	0	0
$163$	1969.09	0.946204	0.473102	−	0.881008i	$-0.343134\pi$
$163$	1969.09	0.946204	0.473102	+	0.881008i	$0.343134\pi$
$164$	0	0
$165$	1619.88	0.764287
$166$	0	0
$167$	1221.66	0.566075	0.283038	−	0.959109i	$-0.408658\pi$
$167$	1221.66	0.566075	0.283038	+	0.959109i	$0.408658\pi$
$168$	0	0
$169$	−1942.91	−0.884347
$170$	0	0
$171$	−1787.21	−0.799249
$172$	0	0
$173$	796.794	0.350168	0.175084	−	0.984553i	$-0.443980\pi$
$173$	796.794	0.350168	0.175084	+	0.984553i	$0.443980\pi$
$174$	0	0
$175$	−2029.05	−0.876468
$176$	0	0
$177$	−4451.99	−1.89058
$178$	0	0
$179$	3114.42	1.30046	0.650230	−	0.759737i	$-0.274672\pi$
$179$	3114.42	1.30046	0.650230	+	0.759737i	$0.274672\pi$
$180$	0	0
$181$	171.535	0.0704425	0.0352213	−	0.999380i	$-0.488786\pi$
$181$	171.535	0.0704425	0.0352213	+	0.999380i	$0.488786\pi$
$182$	0	0
$183$	7305.26	2.95093
$184$	0	0
$185$	1522.98	0.605251
$186$	0	0
$187$	2795.93	1.09336
$188$	0	0
$189$	1241.59	0.477845
$190$	0	0
$191$	3927.65	1.48793	0.743966	−	0.668218i	$-0.232942\pi$
$191$	3927.65	1.48793	0.743966	+	0.668218i	$0.232942\pi$
$192$	0	0
$193$	−3249.02	−1.21176	−0.605880	−	0.795556i	$-0.707179\pi$
$193$	−3249.02	−1.21176	−0.605880	+	0.795556i	$0.707179\pi$
$194$	0	0
$195$	817.179	0.300099
$196$	0	0
$197$	−3423.68	−1.23821	−0.619103	−	0.785309i	$-0.712504\pi$
$197$	−3423.68	−1.23821	−0.619103	+	0.785309i	$0.712504\pi$
$198$	0	0
$199$	−1371.30	−0.488488	−0.244244	−	0.969714i	$-0.578540\pi$
$199$	−1371.30	−0.488488	−0.244244	+	0.969714i	$0.578540\pi$
$200$	0	0
$201$	−1180.33	−0.414198
$202$	0	0
$203$	367.693	0.127128
$204$	0	0
$205$	−2377.80	−0.810112
$206$	0	0
$207$	1608.06	0.539941
$208$	0	0
$209$	−1689.98	−0.559321
$210$	0	0
$211$	2291.73	0.747721	0.373861	−	0.927485i	$-0.378034\pi$
$211$	2291.73	0.747721	0.373861	+	0.927485i	$0.378034\pi$
$212$	0	0
$213$	−1671.54	−0.537708
$214$	0	0
$215$	−935.068	−0.296610
$216$	0	0
$217$	−2413.91	−0.755148
$218$	0	0
$219$	−5196.77	−1.60349
$220$	0	0
$221$	1410.46	0.429312
$222$	0	0
$223$	−419.617	−0.126007	−0.0630036	−	0.998013i	$-0.520068\pi$
$223$	−419.617	−0.126007	−0.0630036	+	0.998013i	$0.520068\pi$
$224$	0	0
$225$	−2723.38	−0.806929
$226$	0	0
$227$	−3017.42	−0.882262	−0.441131	−	0.897443i	$-0.645423\pi$
$227$	−3017.42	−0.882262	−0.441131	+	0.897443i	$0.645423\pi$
$228$	0	0
$229$	2226.56	0.642513	0.321256	−	0.946992i	$-0.395895\pi$
$229$	2226.56	0.642513	0.321256	+	0.946992i	$0.395895\pi$
$230$	0	0
$231$	6114.67	1.74163
$232$	0	0
$233$	1194.86	0.335957	0.167978	−	0.985791i	$-0.446276\pi$
$233$	1194.86	0.335957	0.167978	+	0.985791i	$0.446276\pi$
$234$	0	0
$235$	−1455.42	−0.404006
$236$	0	0
$237$	6396.65	1.75320
$238$	0	0
$239$	4241.03	1.14782	0.573911	−	0.818917i	$-0.305425\pi$
$239$	4241.03	1.14782	0.573911	+	0.818917i	$0.305425\pi$
$240$	0	0
$241$	5571.19	1.48910	0.744548	−	0.667569i	$-0.232665\pi$
$241$	5571.19	1.48910	0.744548	+	0.667569i	$0.232665\pi$
$242$	0	0
$243$	−5346.38	−1.41140
$244$	0	0
$245$	1825.88	0.476126
$246$	0	0
$247$	−852.541	−0.219619
$248$	0	0
$249$	−3593.11	−0.914474
$250$	0	0
$251$	682.681	0.171675	0.0858375	−	0.996309i	$-0.472643\pi$
$251$	682.681	0.171675	0.0858375	+	0.996309i	$0.472643\pi$
$252$	0	0
$253$	1520.57	0.377855
$254$	0	0
$255$	4536.19	1.11399
$256$	0	0
$257$	8093.12	1.96434	0.982169	−	0.188002i	$-0.0602013\pi$
$257$	8093.12	1.96434	0.982169	+	0.188002i	$0.0602013\pi$
$258$	0	0
$259$	5748.89	1.37922
$260$	0	0
$261$	493.516	0.117042
$262$	0	0
$263$	410.300	0.0961984	0.0480992	−	0.998843i	$-0.484684\pi$
$263$	410.300	0.0961984	0.0480992	+	0.998843i	$0.484684\pi$
$264$	0	0
$265$	−1639.32	−0.380009
$266$	0	0
$267$	2042.17	0.468085
$268$	0	0
$269$	6.75940	0.00153207	0.000766037	−	1.00000i	$-0.499756\pi$
$269$	6.75940	0.00153207	0.000766037		1.00000i	$0.499756\pi$
$270$	0	0
$271$	2833.98	0.635247	0.317623	−	0.948217i	$-0.397115\pi$
$271$	2833.98	0.635247	0.317623	+	0.948217i	$0.397115\pi$
$272$	0	0
$273$	3084.67	0.683855
$274$	0	0
$275$	−2575.21	−0.564695
$276$	0	0
$277$	2157.81	0.468052	0.234026	−	0.972230i	$-0.424810\pi$
$277$	2157.81	0.468052	0.234026	+	0.972230i	$0.424810\pi$
$278$	0	0
$279$	−3239.94	−0.695234
$280$	0	0
$281$	−4750.23	−1.00845	−0.504226	−	0.863572i	$-0.668222\pi$
$281$	−4750.23	−1.00845	−0.504226	+	0.863572i	$0.668222\pi$
$282$	0	0
$283$	−910.900	−0.191334	−0.0956668	−	0.995413i	$-0.530498\pi$
$283$	−910.900	−0.191334	−0.0956668	+	0.995413i	$0.530498\pi$
$284$	0	0
$285$	−2741.86	−0.569873
$286$	0	0
$287$	−8975.67	−1.84605
$288$	0	0
$289$	2916.53	0.593635
$290$	0	0
$291$	−1172.14	−0.236124
$292$	0	0
$293$	2026.80	0.404119	0.202060	−	0.979373i	$-0.435237\pi$
$293$	2026.80	0.404119	0.202060	+	0.979373i	$0.435237\pi$
$294$	0	0
$295$	−3777.69	−0.745578
$296$	0	0
$297$	1575.79	0.307868
$298$	0	0
$299$	767.080	0.148366
$300$	0	0
$301$	−3529.67	−0.675903
$302$	0	0
$303$	−5364.10	−1.01703
$304$	0	0
$305$	6198.79	1.16374
$306$	0	0
$307$	326.981	0.0607877	0.0303938	−	0.999538i	$-0.490324\pi$
$307$	326.981	0.0607877	0.0303938	+	0.999538i	$0.490324\pi$
$308$	0	0
$309$	14306.4	2.63386
$310$	0	0
$311$	871.410	0.158885	0.0794423	−	0.996839i	$-0.474686\pi$
$311$	871.410	0.158885	0.0794423	+	0.996839i	$0.474686\pi$
$312$	0	0
$313$	3515.02	0.634762	0.317381	−	0.948298i	$-0.397197\pi$
$313$	3515.02	0.634762	0.317381	+	0.948298i	$0.397197\pi$
$314$	0	0
$315$	5487.07	0.981465
$316$	0	0
$317$	6680.42	1.18363	0.591814	−	0.806075i	$-0.298412\pi$
$317$	6680.42	1.18363	0.591814	+	0.806075i	$0.298412\pi$
$318$	0	0
$319$	466.665	0.0819067
$320$	0	0
$321$	−872.901	−0.151778
$322$	0	0
$323$	−4732.49	−0.815241
$324$	0	0
$325$	−1299.12	−0.221729
$326$	0	0
$327$	−10475.5	−1.77155
$328$	0	0
$329$	−5493.90	−0.920633
$330$	0	0
$331$	−2574.24	−0.427471	−0.213736	−	0.976892i	$-0.568563\pi$
$331$	−2574.24	−0.427471	−0.213736	+	0.976892i	$0.568563\pi$
$332$	0	0
$333$	7716.13	1.26979
$334$	0	0
$335$	−1001.55	−0.163345
$336$	0	0
$337$	−74.0970	−0.0119772	−0.00598861	−	0.999982i	$-0.501906\pi$
$337$	−74.0970	−0.0119772	−0.00598861	+	0.999982i	$0.501906\pi$
$338$	0	0
$339$	5600.04	0.897205
$340$	0	0
$341$	−3063.67	−0.486530
$342$	0	0
$343$	−1647.24	−0.259307
$344$	0	0
$345$	2467.01	0.384984
$346$	0	0
$347$	−2973.72	−0.460050	−0.230025	−	0.973185i	$-0.573881\pi$
$347$	−2973.72	−0.460050	−0.230025	+	0.973185i	$0.573881\pi$
$348$	0	0
$349$	−9351.98	−1.43438	−0.717192	−	0.696875i	$-0.754573\pi$
$349$	−9351.98	−1.43438	−0.717192	+	0.696875i	$0.754573\pi$
$350$	0	0
$351$	794.940	0.120885
$352$	0	0
$353$	2216.90	0.334259	0.167130	−	0.985935i	$-0.446550\pi$
$353$	2216.90	0.334259	0.167130	+	0.985935i	$0.446550\pi$
$354$	0	0
$355$	−1418.36	−0.212053
$356$	0	0
$357$	17123.1	2.53852
$358$	0	0
$359$	2082.23	0.306117	0.153059	−	0.988217i	$-0.451088\pi$
$359$	2082.23	0.306117	0.153059	+	0.988217i	$0.451088\pi$
$360$	0	0
$361$	−3998.49	−0.582955
$362$	0	0
$363$	−2585.00	−0.373766
$364$	0	0
$365$	−4409.66	−0.632362
$366$	0	0
$367$	−4509.22	−0.641360	−0.320680	−	0.947188i	$-0.603911\pi$
$367$	−4509.22	−0.641360	−0.320680	+	0.947188i	$0.603911\pi$
$368$	0	0
$369$	−12047.1	−1.69959
$370$	0	0
$371$	−6188.05	−0.865951
$372$	0	0
$373$	−12249.3	−1.70039	−0.850194	−	0.526470i	$-0.823515\pi$
$373$	−12249.3	−1.70039	−0.850194	+	0.526470i	$0.823515\pi$
$374$	0	0
$375$	−10586.3	−1.45779
$376$	0	0
$377$	235.418	0.0321609
$378$	0	0
$379$	−4981.51	−0.675153	−0.337576	−	0.941298i	$-0.609607\pi$
$379$	−4981.51	−0.675153	−0.337576	+	0.941298i	$0.609607\pi$
$380$	0	0
$381$	20382.9	2.74081
$382$	0	0
$383$	3044.88	0.406229	0.203115	−	0.979155i	$-0.434894\pi$
$383$	3044.88	0.406229	0.203115	+	0.979155i	$0.434894\pi$
$384$	0	0
$385$	5188.54	0.686837
$386$	0	0
$387$	−4737.51	−0.622277
$388$	0	0
$389$	−2733.30	−0.356256	−0.178128	−	0.984007i	$-0.557004\pi$
$389$	−2733.30	−0.356256	−0.178128	+	0.984007i	$0.557004\pi$
$390$	0	0
$391$	4258.09	0.550744
$392$	0	0
$393$	7228.29	0.927784
$394$	0	0
$395$	5427.81	0.691399
$396$	0	0
$397$	950.997	0.120225	0.0601123	−	0.998192i	$-0.480854\pi$
$397$	950.997	0.120225	0.0601123	+	0.998192i	$0.480854\pi$
$398$	0	0
$399$	−10349.9	−1.29861
$400$	0	0
$401$	7606.74	0.947288	0.473644	−	0.880716i	$-0.342938\pi$
$401$	7606.74	0.947288	0.473644	+	0.880716i	$0.342938\pi$
$402$	0	0
$403$	−1545.53	−0.191037
$404$	0	0
$405$	−3394.06	−0.416425
$406$	0	0
$407$	7296.32	0.888612
$408$	0	0
$409$	−4981.58	−0.602257	−0.301129	−	0.953584i	$-0.597363\pi$
$409$	−4981.58	−0.602257	−0.301129	+	0.953584i	$0.597363\pi$
$410$	0	0
$411$	19521.5	2.34288
$412$	0	0
$413$	−14259.9	−1.69899
$414$	0	0
$415$	−3048.89	−0.360637
$416$	0	0
$417$	−11937.9	−1.40192
$418$	0	0
$419$	−4855.53	−0.566129	−0.283065	−	0.959101i	$-0.591351\pi$
$419$	−4855.53	−0.566129	−0.283065	+	0.959101i	$0.591351\pi$
$420$	0	0
$421$	−11276.5	−1.30543	−0.652714	−	0.757604i	$-0.726370\pi$
$421$	−11276.5	−1.30543	−0.652714	+	0.757604i	$0.726370\pi$
$422$	0	0
$423$	−7373.88	−0.847590
$424$	0	0
$425$	−7211.44	−0.823074
$426$	0	0
$427$	23399.0	2.65189
$428$	0	0
$429$	3914.97	0.440598
$430$	0	0
$431$	−4800.16	−0.536463	−0.268232	−	0.963354i	$-0.586439\pi$
$431$	−4800.16	−0.536463	−0.268232	+	0.963354i	$0.586439\pi$
$432$	0	0
$433$	−6242.32	−0.692810	−0.346405	−	0.938085i	$-0.612598\pi$
$433$	−6242.32	−0.692810	−0.346405	+	0.938085i	$0.612598\pi$
$434$	0	0
$435$	757.130	0.0834520
$436$	0	0
$437$	−2573.77	−0.281739
$438$	0	0
$439$	4929.27	0.535903	0.267951	−	0.963432i	$-0.413653\pi$
$439$	4929.27	0.535903	0.267951	+	0.963432i	$0.413653\pi$
$440$	0	0
$441$	9250.77	0.998896
$442$	0	0
$443$	10848.0	1.16344	0.581718	−	0.813390i	$-0.302381\pi$
$443$	10848.0	1.16344	0.581718	+	0.813390i	$0.302381\pi$
$444$	0	0
$445$	1732.86	0.184596
$446$	0	0
$447$	−25113.5	−2.65733
$448$	0	0
$449$	−11515.2	−1.21032	−0.605162	−	0.796102i	$-0.706892\pi$
$449$	−11515.2	−1.21032	−0.605162	+	0.796102i	$0.706892\pi$
$450$	0	0
$451$	−11391.7	−1.18938
$452$	0	0
$453$	−21875.6	−2.26889
$454$	0	0
$455$	2617.46	0.269689
$456$	0	0
$457$	4829.89	0.494383	0.247191	−	0.968967i	$-0.420492\pi$
$457$	4829.89	0.494383	0.247191	+	0.968967i	$0.420492\pi$
$458$	0	0
$459$	4412.74	0.448735
$460$	0	0
$461$	−11689.6	−1.18100	−0.590498	−	0.807039i	$-0.701069\pi$
$461$	−11689.6	−1.18100	−0.590498	+	0.807039i	$0.701069\pi$
$462$	0	0
$463$	5043.86	0.506281	0.253141	−	0.967430i	$-0.418536\pi$
$463$	5043.86	0.506281	0.253141	+	0.967430i	$0.418536\pi$
$464$	0	0
$465$	−4970.57	−0.495709
$466$	0	0
$467$	−17591.1	−1.74308	−0.871542	−	0.490321i	$-0.836880\pi$
$467$	−17591.1	−1.74308	−0.871542	+	0.490321i	$0.836880\pi$
$468$	0	0
$469$	−3780.63	−0.372225
$470$	0	0
$471$	−9962.47	−0.974621
$472$	0	0
$473$	−4479.75	−0.435474
$474$	0	0
$475$	4358.89	0.421052
$476$	0	0
$477$	−8305.58	−0.797246
$478$	0	0
$479$	13059.7	1.24575	0.622875	−	0.782321i	$-0.285964\pi$
$479$	13059.7	1.24575	0.622875	+	0.782321i	$0.285964\pi$
$480$	0	0
$481$	3680.77	0.348916
$482$	0	0
$483$	9312.41	0.877286
$484$	0	0
$485$	−994.605	−0.0931189
$486$	0	0
$487$	−15549.3	−1.44683	−0.723414	−	0.690414i	$-0.757428\pi$
$487$	−15549.3	−1.44683	−0.723414	+	0.690414i	$0.757428\pi$
$488$	0	0
$489$	15305.3	1.41540
$490$	0	0
$491$	−12202.3	−1.12155	−0.560777	−	0.827967i	$-0.689497\pi$
$491$	−12202.3	−1.12155	−0.560777	+	0.827967i	$0.689497\pi$
$492$	0	0
$493$	1306.82	0.119383
$494$	0	0
$495$	6964.03	0.632344
$496$	0	0
$497$	−5354.00	−0.483219
$498$	0	0
$499$	−2450.01	−0.219794	−0.109897	−	0.993943i	$-0.535052\pi$
$499$	−2450.01	−0.219794	−0.109897	+	0.993943i	$0.535052\pi$
$500$	0	0
$501$	9495.65	0.846775
$502$	0	0
$503$	−16579.6	−1.46968	−0.734839	−	0.678241i	$-0.762742\pi$
$503$	−16579.6	−1.46968	−0.734839	+	0.678241i	$0.762742\pi$
$504$	0	0
$505$	−4551.65	−0.401081
$506$	0	0
$507$	−15101.8	−1.32287
$508$	0	0
$509$	−11074.6	−0.964387	−0.482193	−	0.876065i	$-0.660160\pi$
$509$	−11074.6	−0.964387	−0.482193	+	0.876065i	$0.660160\pi$
$510$	0	0
$511$	−16645.5	−1.44100
$512$	0	0
$513$	−2667.24	−0.229555
$514$	0	0
$515$	12139.5	1.03870
$516$	0	0
$517$	−6972.69	−0.593150
$518$	0	0
$519$	6193.30	0.523807
$520$	0	0
$521$	−3400.02	−0.285907	−0.142953	−	0.989729i	$-0.545660\pi$
$521$	−3400.02	−0.285907	−0.142953	+	0.989729i	$0.545660\pi$
$522$	0	0
$523$	−2856.01	−0.238785	−0.119393	−	0.992847i	$-0.538095\pi$
$523$	−2856.01	−0.238785	−0.119393	+	0.992847i	$0.538095\pi$
$524$	0	0
$525$	−15771.4	−1.31108
$526$	0	0
$527$	−8579.28	−0.709144
$528$	0	0
$529$	−9851.23	−0.809668
$530$	0	0
$531$	−19139.6	−1.56420
$532$	0	0
$533$	−5746.74	−0.467015
$534$	0	0
$535$	−740.691	−0.0598558
$536$	0	0
$537$	24207.7	1.94532
$538$	0	0
$539$	8747.47	0.699035
$540$	0	0
$541$	−18996.6	−1.50966	−0.754829	−	0.655921i	$-0.772280\pi$
$541$	−18996.6	−1.50966	−0.754829	+	0.655921i	$0.772280\pi$
$542$	0	0
$543$	1333.30	0.105373
$544$	0	0
$545$	−8888.86	−0.698637
$546$	0	0
$547$	−7603.46	−0.594333	−0.297167	−	0.954826i	$-0.596042\pi$
$547$	−7603.46	−0.594333	−0.297167	+	0.954826i	$0.596042\pi$
$548$	0	0
$549$	31406.1	2.44149
$550$	0	0
$551$	−789.894	−0.0610719
$552$	0	0
$553$	20488.7	1.57553
$554$	0	0
$555$	11837.7	0.905377
$556$	0	0
$557$	2936.30	0.223367	0.111683	−	0.993744i	$-0.464376\pi$
$557$	2936.30	0.223367	0.111683	+	0.993744i	$0.464376\pi$
$558$	0	0
$559$	−2259.90	−0.170990
$560$	0	0
$561$	21732.1	1.63553
$562$	0	0
$563$	23536.9	1.76192	0.880961	−	0.473188i	$-0.156897\pi$
$563$	23536.9	1.76192	0.880961	+	0.473188i	$0.156897\pi$
$564$	0	0
$565$	4751.85	0.353826
$566$	0	0
$567$	−12811.8	−0.948934
$568$	0	0
$569$	−5659.60	−0.416982	−0.208491	−	0.978024i	$-0.566855\pi$
$569$	−5659.60	−0.416982	−0.208491	+	0.978024i	$0.566855\pi$
$570$	0	0
$571$	−6891.00	−0.505043	−0.252521	−	0.967591i	$-0.581260\pi$
$571$	−6891.00	−0.505043	−0.252521	+	0.967591i	$0.581260\pi$
$572$	0	0
$573$	30528.8	2.22575
$574$	0	0
$575$	−3921.95	−0.284446
$576$	0	0
$577$	10652.2	0.768556	0.384278	−	0.923217i	$-0.374450\pi$
$577$	10652.2	0.768556	0.384278	+	0.923217i	$0.374450\pi$
$578$	0	0
$579$	−25253.9	−1.81264
$580$	0	0
$581$	−11508.9	−0.821805
$582$	0	0
$583$	−7853.69	−0.557919
$584$	0	0
$585$	3513.14	0.248291
$586$	0	0
$587$	14006.3	0.984841	0.492421	−	0.870357i	$-0.336112\pi$
$587$	14006.3	0.984841	0.492421	+	0.870357i	$0.336112\pi$
$588$	0	0
$589$	5185.67	0.362770
$590$	0	0
$591$	−26611.5	−1.85220
$592$	0	0
$593$	3528.04	0.244316	0.122158	−	0.992511i	$-0.461019\pi$
$593$	3528.04	0.244316	0.122158	+	0.992511i	$0.461019\pi$
$594$	0	0
$595$	14529.6	1.00110
$596$	0	0
$597$	−10658.8	−0.730715
$598$	0	0
$599$	19024.9	1.29772	0.648861	−	0.760907i	$-0.275246\pi$
$599$	19024.9	1.29772	0.648861	+	0.760907i	$0.275246\pi$
$600$	0	0
$601$	−1065.92	−0.0723460	−0.0361730	−	0.999346i	$-0.511517\pi$
$601$	−1065.92	−0.0723460	−0.0361730	+	0.999346i	$0.511517\pi$
$602$	0	0
$603$	−5074.35	−0.342692
$604$	0	0
$605$	−2193.47	−0.147400
$606$	0	0
$607$	12909.7	0.863242	0.431621	−	0.902055i	$-0.357942\pi$
$607$	12909.7	0.863242	0.431621	+	0.902055i	$0.357942\pi$
$608$	0	0
$609$	2858.00	0.190167
$610$	0	0
$611$	−3517.51	−0.232902
$612$	0	0
$613$	12393.3	0.816578	0.408289	−	0.912853i	$-0.366126\pi$
$613$	12393.3	0.816578	0.408289	+	0.912853i	$0.366126\pi$
$614$	0	0
$615$	−18482.1	−1.21182
$616$	0	0
$617$	−18921.2	−1.23459	−0.617293	−	0.786733i	$-0.711771\pi$
$617$	−18921.2	−1.23459	−0.617293	+	0.786733i	$0.711771\pi$
$618$	0	0
$619$	−21378.2	−1.38814	−0.694072	−	0.719906i	$-0.744185\pi$
$619$	−21378.2	−1.38814	−0.694072	+	0.719906i	$0.744185\pi$
$620$	0	0
$621$	2399.87	0.155078
$622$	0	0
$623$	6541.15	0.420651
$624$	0	0
$625$	1204.57	0.0770926
$626$	0	0
$627$	−13135.8	−0.836672
$628$	0	0
$629$	20432.1	1.29520
$630$	0	0
$631$	−9602.80	−0.605834	−0.302917	−	0.953017i	$-0.597961\pi$
$631$	−9602.80	−0.605834	−0.302917	+	0.953017i	$0.597961\pi$
$632$	0	0
$633$	17813.1	1.11849
$634$	0	0
$635$	17295.7	1.08088
$636$	0	0
$637$	4412.83	0.274478
$638$	0	0
$639$	−7186.12	−0.444880
$640$	0	0
$641$	4450.84	0.274256	0.137128	−	0.990553i	$-0.456213\pi$
$641$	4450.84	0.274256	0.137128	+	0.990553i	$0.456213\pi$
$642$	0	0
$643$	9180.04	0.563026	0.281513	−	0.959557i	$-0.409164\pi$
$643$	9180.04	0.563026	0.281513	+	0.959557i	$0.409164\pi$
$644$	0	0
$645$	−7268.07	−0.443690
$646$	0	0
$647$	5546.17	0.337005	0.168503	−	0.985701i	$-0.446107\pi$
$647$	5546.17	0.337005	0.168503	+	0.985701i	$0.446107\pi$
$648$	0	0
$649$	−18098.3	−1.09464
$650$	0	0
$651$	−18762.8	−1.12960
$652$	0	0
$653$	−8948.56	−0.536270	−0.268135	−	0.963381i	$-0.586407\pi$
$653$	−8948.56	−0.536270	−0.268135	+	0.963381i	$0.586407\pi$
$654$	0	0
$655$	6133.49	0.365886
$656$	0	0
$657$	−22341.5	−1.32667
$658$	0	0
$659$	−21404.4	−1.26525	−0.632623	−	0.774460i	$-0.718022\pi$
$659$	−21404.4	−1.26525	−0.632623	+	0.774460i	$0.718022\pi$
$660$	0	0
$661$	−33177.6	−1.95228	−0.976140	−	0.217140i	$-0.930327\pi$
$661$	−33177.6	−1.95228	−0.976140	+	0.217140i	$0.930327\pi$
$662$	0	0
$663$	10963.2	0.642195
$664$	0	0
$665$	−8782.30	−0.512125
$666$	0	0
$667$	710.713	0.0412577
$668$	0	0
$669$	−3261.59	−0.188491
$670$	0	0
$671$	29697.4	1.70858
$672$	0	0
$673$	30638.5	1.75487	0.877436	−	0.479694i	$-0.159252\pi$
$673$	30638.5	1.75487	0.877436	+	0.479694i	$0.159252\pi$
$674$	0	0
$675$	−4064.39	−0.231760
$676$	0	0
$677$	17633.7	1.00106	0.500529	−	0.865720i	$-0.333139\pi$
$677$	17633.7	1.00106	0.500529	+	0.865720i	$0.333139\pi$
$678$	0	0
$679$	−3754.41	−0.212196
$680$	0	0
$681$	−23453.8	−1.31975
$682$	0	0
$683$	19570.6	1.09641	0.548206	−	0.836344i	$-0.315311\pi$
$683$	19570.6	1.09641	0.548206	+	0.836344i	$0.315311\pi$
$684$	0	0
$685$	16564.8	0.923952
$686$	0	0
$687$	17306.6	0.961116
$688$	0	0
$689$	−3961.95	−0.219068
$690$	0	0
$691$	149.923	0.00825375	0.00412688	−	0.999991i	$-0.498686\pi$
$691$	149.923	0.00825375	0.00412688	+	0.999991i	$0.498686\pi$
$692$	0	0
$693$	26287.7	1.44096
$694$	0	0
$695$	−10129.8	−0.552870
$696$	0	0
$697$	−31900.4	−1.73359
$698$	0	0
$699$	9287.38	0.502548
$700$	0	0
$701$	11086.5	0.597335	0.298667	−	0.954357i	$-0.403458\pi$
$701$	11086.5	0.597335	0.298667	+	0.954357i	$0.403458\pi$
$702$	0	0
$703$	−12350.0	−0.662574
$704$	0	0
$705$	−11312.7	−0.604341
$706$	0	0
$707$	−17181.5	−0.913968
$708$	0	0
$709$	−3858.40	−0.204380	−0.102190	−	0.994765i	$-0.532585\pi$
$709$	−3858.40	−0.204380	−0.102190	+	0.994765i	$0.532585\pi$
$710$	0	0
$711$	27499.9	1.45053
$712$	0	0
$713$	−4665.84	−0.245073
$714$	0	0
$715$	3322.00	0.173756
$716$	0	0
$717$	32964.6	1.71699
$718$	0	0
$719$	32717.8	1.69704	0.848519	−	0.529166i	$-0.177495\pi$
$719$	32717.8	1.69704	0.848519	+	0.529166i	$0.177495\pi$
$720$	0	0
$721$	45824.0	2.36696
$722$	0	0
$723$	43303.6	2.22750
$724$	0	0
$725$	−1203.65	−0.0616587
$726$	0	0
$727$	25847.2	1.31859	0.659297	−	0.751883i	$-0.270854\pi$
$727$	25847.2	1.31859	0.659297	+	0.751883i	$0.270854\pi$
$728$	0	0
$729$	−27662.0	−1.40537
$730$	0	0
$731$	−12544.8	−0.634727
$732$	0	0
$733$	14328.7	0.722023	0.361011	−	0.932561i	$-0.382432\pi$
$733$	14328.7	0.722023	0.361011	+	0.932561i	$0.382432\pi$
$734$	0	0
$735$	14192.1	0.712223
$736$	0	0
$737$	−4798.27	−0.239819
$738$	0	0
$739$	27101.5	1.34905	0.674524	−	0.738253i	$-0.264349\pi$
$739$	27101.5	1.34905	0.674524	+	0.738253i	$0.264349\pi$
$740$	0	0
$741$	−6626.61	−0.328522
$742$	0	0
$743$	23322.1	1.15155	0.575777	−	0.817607i	$-0.304699\pi$
$743$	23322.1	1.15155	0.575777	+	0.817607i	$0.304699\pi$
$744$	0	0
$745$	−21309.8	−1.04796
$746$	0	0
$747$	−15447.2	−0.756603
$748$	0	0
$749$	−2795.94	−0.136397
$750$	0	0
$751$	−25994.0	−1.26303	−0.631515	−	0.775364i	$-0.717566\pi$
$751$	−25994.0	−1.26303	−0.631515	+	0.775364i	$0.717566\pi$
$752$	0	0
$753$	5306.32	0.256804
$754$	0	0
$755$	−18562.3	−0.894770
$756$	0	0
$757$	31317.8	1.50365	0.751825	−	0.659363i	$-0.229174\pi$
$757$	31317.8	1.50365	0.751825	+	0.659363i	$0.229174\pi$
$758$	0	0
$759$	11819.0	0.565222
$760$	0	0
$761$	−16497.5	−0.785853	−0.392926	−	0.919570i	$-0.628537\pi$
$761$	−16497.5	−0.785853	−0.392926	+	0.919570i	$0.628537\pi$
$762$	0	0
$763$	−33553.4	−1.59203
$764$	0	0
$765$	19501.6	0.921675
$766$	0	0
$767$	−9130.02	−0.429812
$768$	0	0
$769$	−24867.3	−1.16611	−0.583055	−	0.812433i	$-0.698143\pi$
$769$	−24867.3	−1.16611	−0.583055	+	0.812433i	$0.698143\pi$
$770$	0	0
$771$	62905.9	2.93839
$772$	0	0
$773$	−2661.15	−0.123823	−0.0619114	−	0.998082i	$-0.519720\pi$
$773$	−2661.15	−0.123823	−0.0619114	+	0.998082i	$0.519720\pi$
$774$	0	0
$775$	7902.00	0.366256
$776$	0	0
$777$	44684.8	2.06314
$778$	0	0
$779$	19281.9	0.886837
$780$	0	0
$781$	−6795.14	−0.311331
$782$	0	0
$783$	736.525	0.0336159
$784$	0	0
$785$	−8453.54	−0.384356
$786$	0	0
$787$	11018.1	0.499049	0.249524	−	0.968369i	$-0.419726\pi$
$787$	11018.1	0.499049	0.249524	+	0.968369i	$0.419726\pi$
$788$	0	0
$789$	3189.17	0.143900
$790$	0	0
$791$	17937.2	0.806286
$792$	0	0
$793$	14981.4	0.670877
$794$	0	0
$795$	−12742.0	−0.568445
$796$	0	0
$797$	−5953.61	−0.264602	−0.132301	−	0.991210i	$-0.542237\pi$
$797$	−5953.61	−0.264602	−0.132301	+	0.991210i	$0.542237\pi$
$798$	0	0
$799$	−19525.8	−0.864549
$800$	0	0
$801$	8779.50	0.387276
$802$	0	0
$803$	−21126.0	−0.928417
$804$	0	0
$805$	7901.94	0.345971
$806$	0	0
$807$	52.5393	0.00229178
$808$	0	0
$809$	27554.3	1.19747	0.598737	−	0.800946i	$-0.295670\pi$
$809$	27554.3	1.19747	0.598737	+	0.800946i	$0.295670\pi$
$810$	0	0
$811$	4817.57	0.208592	0.104296	−	0.994546i	$-0.466741\pi$
$811$	4817.57	0.208592	0.104296	+	0.994546i	$0.466741\pi$
$812$	0	0
$813$	22027.9	0.950247
$814$	0	0
$815$	12987.1	0.558184
$816$	0	0
$817$	7582.59	0.324701
$818$	0	0
$819$	13261.3	0.565797
$820$	0	0
$821$	−13914.6	−0.591504	−0.295752	−	0.955265i	$-0.595570\pi$
$821$	−13914.6	−0.591504	−0.295752	+	0.955265i	$0.595570\pi$
$822$	0	0
$823$	−36653.5	−1.55244	−0.776221	−	0.630461i	$-0.782866\pi$
$823$	−36653.5	−1.55244	−0.776221	+	0.630461i	$0.782866\pi$
$824$	0	0
$825$	−20016.5	−0.844711
$826$	0	0
$827$	31428.3	1.32149	0.660743	−	0.750612i	$-0.270241\pi$
$827$	31428.3	1.32149	0.660743	+	0.750612i	$0.270241\pi$
$828$	0	0
$829$	20810.9	0.871885	0.435942	−	0.899975i	$-0.356415\pi$
$829$	20810.9	0.871885	0.435942	+	0.899975i	$0.356415\pi$
$830$	0	0
$831$	16772.2	0.700145
$832$	0	0
$833$	24495.8	1.01888
$834$	0	0
$835$	8057.43	0.333939
$836$	0	0
$837$	−4835.30	−0.199680
$838$	0	0
$839$	11010.0	0.453050	0.226525	−	0.974005i	$-0.427263\pi$
$839$	11010.0	0.453050	0.226525	+	0.974005i	$0.427263\pi$
$840$	0	0
$841$	−24170.9	−0.991057
$842$	0	0
$843$	−36922.5	−1.50852
$844$	0	0
$845$	−12814.5	−0.521694
$846$	0	0
$847$	−8279.85	−0.335890
$848$	0	0
$849$	−7080.22	−0.286210
$850$	0	0
$851$	11112.0	0.447608
$852$	0	0
$853$	18116.0	0.727175	0.363587	−	0.931560i	$-0.381552\pi$
$853$	18116.0	0.727175	0.363587	+	0.931560i	$0.381552\pi$
$854$	0	0
$855$	−11787.6	−0.471493
$856$	0	0
$857$	38510.8	1.53501	0.767505	−	0.641043i	$-0.221498\pi$
$857$	38510.8	1.53501	0.767505	+	0.641043i	$0.221498\pi$
$858$	0	0
$859$	−32858.7	−1.30515	−0.652576	−	0.757723i	$-0.726312\pi$
$859$	−32858.7	−1.30515	−0.652576	+	0.757723i	$0.726312\pi$
$860$	0	0
$861$	−69765.8	−2.76146
$862$	0	0
$863$	−22079.5	−0.870911	−0.435456	−	0.900210i	$-0.643413\pi$
$863$	−22079.5	−0.870911	−0.435456	+	0.900210i	$0.643413\pi$
$864$	0	0
$865$	5255.26	0.206571
$866$	0	0
$867$	22669.5	0.888001
$868$	0	0
$869$	26003.7	1.01509
$870$	0	0
$871$	−2420.58	−0.0941656
$872$	0	0
$873$	−5039.15	−0.195360
$874$	0	0
$875$	−33908.2	−1.31007
$876$	0	0
$877$	−5773.00	−0.222281	−0.111140	−	0.993805i	$-0.535450\pi$
$877$	−5773.00	−0.222281	−0.111140	+	0.993805i	$0.535450\pi$
$878$	0	0
$879$	15753.9	0.604510
$880$	0	0
$881$	−7132.59	−0.272762	−0.136381	−	0.990656i	$-0.543547\pi$
$881$	−7132.59	−0.272762	−0.136381	+	0.990656i	$0.543547\pi$
$882$	0	0
$883$	−27110.4	−1.03323	−0.516613	−	0.856219i	$-0.672807\pi$
$883$	−27110.4	−1.03323	−0.516613	+	0.856219i	$0.672807\pi$
$884$	0	0
$885$	−29363.1	−1.11529
$886$	0	0
$887$	45045.7	1.70517	0.852585	−	0.522589i	$-0.175034\pi$
$887$	45045.7	1.70517	0.852585	+	0.522589i	$0.175034\pi$
$888$	0	0
$889$	65287.3	2.46306
$890$	0	0
$891$	−16260.4	−0.611384
$892$	0	0
$893$	11802.2	0.442269
$894$	0	0
$895$	20541.1	0.767167
$896$	0	0
$897$	5962.34	0.221936
$898$	0	0
$899$	−1431.96	−0.0531239
$900$	0	0
$901$	−21992.9	−0.813197
$902$	0	0
$903$	−27435.3	−1.01106
$904$	0	0
$905$	1131.36	0.0415554
$906$	0	0
$907$	−47599.9	−1.74259	−0.871295	−	0.490760i	$-0.836719\pi$
$907$	−47599.9	−1.74259	−0.871295	+	0.490760i	$0.836719\pi$
$908$	0	0
$909$	−23060.9	−0.841453
$910$	0	0
$911$	42503.8	1.54579	0.772895	−	0.634534i	$-0.218808\pi$
$911$	42503.8	1.54579	0.772895	+	0.634534i	$0.218808\pi$
$912$	0	0
$913$	−14606.7	−0.529477
$914$	0	0
$915$	48181.8	1.74081
$916$	0	0
$917$	23152.5	0.833766
$918$	0	0
$919$	8819.41	0.316568	0.158284	−	0.987394i	$-0.449404\pi$
$919$	8819.41	0.316568	0.158284	+	0.987394i	$0.449404\pi$
$920$	0	0
$921$	2541.55	0.0909305
$922$	0	0
$923$	−3427.94	−0.122245
$924$	0	0
$925$	−18819.1	−0.668940
$926$	0	0
$927$	61504.8	2.17916
$928$	0	0
$929$	−14155.6	−0.499925	−0.249963	−	0.968256i	$-0.580418\pi$
$929$	−14155.6	−0.499925	−0.249963	+	0.968256i	$0.580418\pi$
$930$	0	0
$931$	−14806.3	−0.521220
$932$	0	0
$933$	6773.27	0.237671
$934$	0	0
$935$	18440.6	0.644996
$936$	0	0
$937$	38518.7	1.34296	0.671479	−	0.741023i	$-0.265659\pi$
$937$	38518.7	1.34296	0.671479	+	0.741023i	$0.265659\pi$
$938$	0	0
$939$	27321.4	0.949523
$940$	0	0
$941$	−39595.4	−1.37170	−0.685852	−	0.727741i	$-0.740570\pi$
$941$	−39595.4	−1.37170	−0.685852	+	0.727741i	$0.740570\pi$
$942$	0	0
$943$	−17349.0	−0.599112
$944$	0	0
$945$	8188.93	0.281890
$946$	0	0
$947$	46442.1	1.59363	0.796814	−	0.604225i	$-0.206517\pi$
$947$	46442.1	1.59363	0.796814	+	0.604225i	$0.206517\pi$
$948$	0	0
$949$	−10657.4	−0.364545
$950$	0	0
$951$	51925.4	1.77055
$952$	0	0
$953$	−20600.1	−0.700211	−0.350106	−	0.936710i	$-0.613854\pi$
$953$	−20600.1	−0.700211	−0.350106	+	0.936710i	$0.613854\pi$
$954$	0	0
$955$	25904.8	0.877760
$956$	0	0
$957$	3627.28	0.122522
$958$	0	0
$959$	62528.2	2.10547
$960$	0	0
$961$	−20390.2	−0.684441
$962$	0	0
$963$	−3752.70	−0.125575
$964$	0	0
$965$	−21428.9	−0.714841
$966$	0	0
$967$	38210.9	1.27071	0.635356	−	0.772219i	$-0.280853\pi$
$967$	38210.9	1.27071	0.635356	+	0.772219i	$0.280853\pi$
$968$	0	0
$969$	−36784.6	−1.21950
$970$	0	0
$971$	52643.9	1.73988	0.869941	−	0.493157i	$-0.164157\pi$
$971$	52643.9	1.73988	0.869941	+	0.493157i	$0.164157\pi$
$972$	0	0
$973$	−38237.6	−1.25986
$974$	0	0
$975$	−10097.7	−0.331678
$976$	0	0
$977$	−7985.95	−0.261508	−0.130754	−	0.991415i	$-0.541740\pi$
$977$	−7985.95	−0.261508	−0.130754	+	0.991415i	$0.541740\pi$
$978$	0	0
$979$	8301.83	0.271019
$980$	0	0
$981$	−45035.3	−1.46571
$982$	0	0
$983$	10703.1	0.347279	0.173639	−	0.984809i	$-0.444447\pi$
$983$	10703.1	0.347279	0.173639	+	0.984809i	$0.444447\pi$
$984$	0	0
$985$	−22580.9	−0.730442
$986$	0	0
$987$	−42702.8	−1.37715
$988$	0	0
$989$	−6822.49	−0.219355
$990$	0	0
$991$	−23945.4	−0.767558	−0.383779	−	0.923425i	$-0.625377\pi$
$991$	−23945.4	−0.767558	−0.383779	+	0.923425i	$0.625377\pi$
$992$	0	0
$993$	−20009.0	−0.639442
$994$	0	0
$995$	−9044.42	−0.288168
$996$	0	0
$997$	20212.7	0.642068	0.321034	−	0.947068i	$-0.395970\pi$
$997$	20212.7	0.642068	0.321034	+	0.947068i	$0.395970\pi$
$998$	0	0
$999$	11515.6	0.364702

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1024.4.a.n.1.9		10
4.3	odd	2		1024.4.a.m.1.2		10
8.3	odd	2		1024.4.a.m.1.9		10
8.5	even	2	inner	1024.4.a.n.1.2		10
16.3	odd	4		1024.4.b.k.513.2		10
16.5	even	4		1024.4.b.j.513.2		10
16.11	odd	4		1024.4.b.k.513.9		10
16.13	even	4		1024.4.b.j.513.9		10
32.3	odd	8		64.4.e.a.17.1		10
32.5	even	8		128.4.e.b.97.1		10
32.11	odd	8		64.4.e.a.49.1		10
32.13	even	8		128.4.e.b.33.1		10
32.19	odd	8		128.4.e.a.33.5		10
32.21	even	8		16.4.e.a.5.1	✓	10
32.27	odd	8		128.4.e.a.97.5		10
32.29	even	8		16.4.e.a.13.1	yes	10
96.11	even	8		576.4.k.a.433.4		10
96.29	odd	8		144.4.k.a.109.5		10
96.35	even	8		576.4.k.a.145.4		10
96.53	odd	8		144.4.k.a.37.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.4.e.a.5.1	✓	10	32.21	even	8
16.4.e.a.13.1	yes	10	32.29	even	8
64.4.e.a.17.1		10	32.3	odd	8
64.4.e.a.49.1		10	32.11	odd	8
128.4.e.a.33.5		10	32.19	odd	8
128.4.e.a.97.5		10	32.27	odd	8
128.4.e.b.33.1		10	32.13	even	8
128.4.e.b.97.1		10	32.5	even	8
144.4.k.a.37.5		10	96.53	odd	8
144.4.k.a.109.5		10	96.29	odd	8
576.4.k.a.145.4		10	96.35	even	8
576.4.k.a.433.4		10	96.11	even	8
1024.4.a.m.1.2		10	4.3	odd	2
1024.4.a.m.1.9		10	8.3	odd	2
1024.4.a.n.1.2		10	8.5	even	2	inner
1024.4.a.n.1.9		10	1.1	even	1	trivial
1024.4.b.j.513.2		10	16.5	even	4
1024.4.b.j.513.9		10	16.13	even	4
1024.4.b.k.513.2		10	16.3	odd	4
1024.4.b.k.513.9		10	16.11	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1024.a (trivial)

\(f(q)\)	\(=\)	\(q+7.77277 q^{3} +6.59550 q^{5} +24.8965 q^{7} +33.4160 q^{9} +O(q^{10})\)	\(q+7.77277 q^{3} +6.59550 q^{5} +24.8965 q^{7} +33.4160 q^{9} +31.5979 q^{11} +15.9402 q^{13} +51.2653 q^{15} +88.4846 q^{17} -53.4838 q^{19} +193.515 q^{21} +48.1224 q^{23} -81.4994 q^{25} +49.8702 q^{27} +14.7689 q^{29} -96.9578 q^{31} +245.604 q^{33} +164.205 q^{35} +230.911 q^{37} +123.899 q^{39} -360.519 q^{41} -141.774 q^{43} +220.395 q^{45} -220.669 q^{47} +276.837 q^{49} +687.771 q^{51} -248.551 q^{53} +208.404 q^{55} -415.717 q^{57} -572.767 q^{59} +939.852 q^{61} +831.942 q^{63} +105.133 q^{65} -151.854 q^{67} +374.045 q^{69} -215.050 q^{71} -668.587 q^{73} -633.476 q^{75} +786.679 q^{77} +822.956 q^{79} -514.603 q^{81} -462.269 q^{83} +583.600 q^{85} +114.795 q^{87} +262.733 q^{89} +396.855 q^{91} -753.631 q^{93} -352.752 q^{95} -150.801 q^{97} +1055.88 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 28 q^{7} + 54 q^{9}+O(q^{10}) \) 10 * q + 28 * q^7 + 54 * q^9	\( 10 q + 28 q^{7} + 54 q^{9} + 124 q^{15} + 4 q^{17} + 276 q^{23} + 50 q^{25} + 368 q^{31} - 4 q^{33} + 732 q^{39} + 944 q^{47} - 94 q^{49} + 1380 q^{55} + 108 q^{57} + 2628 q^{63} - 492 q^{65} + 3468 q^{71} - 296 q^{73} + 4416 q^{79} - 482 q^{81} + 6036 q^{87} + 88 q^{89} + 6900 q^{95} - 4 q^{97}+O(q^{100}) \) 10 * q + 28 * q^7 + 54 * q^9 + 124 * q^15 + 4 * q^17 + 276 * q^23 + 50 * q^25 + 368 * q^31 - 4 * q^33 + 732 * q^39 + 944 * q^47 - 94 * q^49 + 1380 * q^55 + 108 * q^57 + 2628 * q^63 - 492 * q^65 + 3468 * q^71 - 296 * q^73 + 4416 * q^79 - 482 * q^81 + 6036 * q^87 + 88 * q^89 + 6900 * q^95 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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